Skip to main content
SpringerLink
Log in

Distributed Source Models: Standard Solutions and New Developments

  • Chapter
Analysis of Neurophysiological Brain Functioning

Part of the book series: Springer Series in Synergetics ((SSSYN))

Abstract

When solving the electro-magnetic inverse problem we face the solution of a linear inverse problem with discrete measurements (Bertero et al. 1985, 1988) that can be written as:

$${d_i} = \int_R {{L_i}(x) \circ j(x)dR + {n_i}{\text{ }}i = 1..{N_s},}$$
((1))

where L i (x) stands for the vector lead field associated to the ith sensor, that is operated on (scalar or vector product ○) with the unknown current density vector j(x). The integration of this product over the whole region R results in the ith measurement. The n i stands for the noise contribution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Allasia, G. (1995): A class of interpolating positive linear operators: Theoretical and Computational aspects. In: Singh, S.P., Carbone, A. and Watson, B. (eds.): Approximation theory, Wavelets and Applications. ( Kluwer Academy Publishers, Dordrecht/Boston/London ).

    Google Scholar 

  • Achim, A., Richer, F. and Saint-Hilaire, J. (1991): Methodological considerations for the evaluation of spatio-temporal source models. Electroenceph. clin. Neurophysiol., 1991, 79: 227 – 240.

    Google Scholar 

  • Backus, G.E. and Gilbert, J.F. (1967): Numerical applications of a formalism for geophysical inverse problems. Geophys. J. Roy. Astron. Soc. 13: 247 – 276.

    ADS  Google Scholar 

  • Backus, G.E. and Gilbert, J.F. (1968): The resolving power of gross earth data. Geophys. J. Roy. Astron. Soc. 16: 169 – 205.

    ADS  MATH  Google Scholar 

  • Backus, G.E. and Gilbert, J.F. (1970): Uniqueness in the inversion of gross earth data. Phil. Trans. R. Soc. 266: 123 – 192.

    Article  MathSciNet  ADS  Google Scholar 

  • Baillet, S. and Garnero, L. (1997): A bayesian approach to introducing anatomo-functional priors in the EEG/MEG inverse problem. IEEE Trans. Biomed. Eng., 44: 374 – 385.

    Article  Google Scholar 

  • Ben-Israel, A., and Greville T.N.E. (1974)\Generalized inverses: Theory and applications. (John Wiley and Sons, Inc. New York).

    Google Scholar 

  • Bertero, M., De Mol, C. and Pike, E.R. (1985): Linear inverse problems with discrete data. I: General formulation and singular system analysis. Inverse Problem, 1: 301 – 330.

    Article  ADS  MATH  Google Scholar 

  • Bertero, M., De Mol, G., and Pike, E.R. (1988) Linear inverse problems with discrete data. II. Stability and regularization. Inverse Problem 4: 573 – 594.

    Article  ADS  MATH  Google Scholar 

  • Cabrera Fernandez, D., Grave de Peralta, R. and Gonzalez Andino, S. (1995). Some limitations of spatio temporal source models. Brain Topography 7 (3): 233 – 243.

    Article  Google Scholar 

  • Clarke, C.J.S., Ioannides, A.A. and Bolton, J.P.R. (1989): Localized and distributed source solutions for the biomagnetic inverse problem I. In: Williamson S J, Hoke M, Stroink G and Kotani M (eds): Advances in Biomagnetism. New York: Plenum Press, pp. 587 – 590.

    Google Scholar 

  • Dale, A.M. and Sereno, M.I. (1993): Improved localization of cortical activity by combining EEG and MEG with MRI cortical surface reconstruction: A linear approach. J. Cogn. Neurosci. 5, 162: 176.

    Google Scholar 

  • Fuchs, M., Wischmann, H.A. and Wagner, M. (1994): Generalized minimum norm least squares reconstruction algorithms. In: ISBET Newsletter No. 5, November 1994. Ed: W. Skrandies. 8 – 11.

    Google Scholar 

  • Gonzalez Andino, S.L., Grave de Peralta Menenedez, R., Biscay Lirio, R., Jimenez Sobrino, J.C., Pascual Marqui, R.D., Lemagne, J. and Valdes Sosa, P.A.,. (1989): Projective methods for the magnetic direct problem, in: Advances in Biomagnetism, edited by S.J. Williamson, M.Hoke, G.Stroink and M. Kotani. Plenum Press, New York, pp. 615 – 618.

    Google Scholar 

  • Goronidnitsky, I.F. and Rao, B.D. (1997): Spatial signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans. Sig. Proc., 45 (3): 1 – 16.

    Google Scholar 

  • Grave de Peralta Menendez, R. and Gonzalez Andino, S.L. (1994): Single Dipole Localization: Some numerical aspects and a practical rejection criterion for the fitted parameters. Brain Topography 6 (4): 277 – 282.

    Article  Google Scholar 

  • Grave de Peralta Menendez, R. and Gonzalez Andino S.L. (1998): A critical analysis of linear inverse solutions. IEEE Trans. Biomed. Engn. 45: 440 – 448

    Article  Google Scholar 

  • Grave de Peralta Menendez, R., Gonzalez Andino, S.L., Cabrera Fernandez, D., Tor- rez, L. (1993b): A proposal for the inverse solution combining different sources of information. IV International Symposium of the ISBET Society. Havana, Cuba. Abstract published in Brain Topography.

    Google Scholar 

  • Grave de Peralta Menendez, R., Oropesa Farras, E., Gonzalez Andino, S.L., Cabrera Fernandez, D., Aubert Vazquez, E. (1993a): An approach to phisiollogically meaningful distributed inverse solutions to the neuroelectromagnetic inverse problem. Technical Report 1. July, 1993. Cuban Neuroscience Center, Havana, Cuba.

    Google Scholar 

  • Grave de Peralta Menendez, R., Gonzalez Andino, S. and Lütkenhöner, B. (1996): Figures of merit to compare linear distributed inverse solutions. Brain Topography. Vol. 9. No. 2: 117 – 124.

    Article  Google Scholar 

  • Grave de Peralta Menendez, R., Hauk, O., Gonzalez Andino, S., Vogt, H. and Michel, C.M. (1997a) Linear inverse solutions with optimal resolution kernels applied to the electromagnetic tomography. Human Brain Mapping, 5: 454 – 467

    Article  Google Scholar 

  • Grave de Peralta Menendez, E., Gonzalez Andino, S., Hauk, O., Spinelli, L., Michel, C.M. (1997b): A linear inverse solution with optimal resolution properties: WROP. Biomedizinische Technik, 42: 53 – 56

    Article  Google Scholar 

  • Grave de Peralta Menendez, R., Gonzalez Andino, S.L., Morand, S., Michel, C.M., (1997c): Imaging the electrical activity of the brain: ELECTRA. submitted.

    Google Scholar 

  • Greenblatt, R.E. (1993): Probabilistic reconstruction of multiple sources in the neuroelectromagnetic inverse problem. Inverse Problems 9: 271 – 284.

    Article  ADS  MATH  Google Scholar 

  • Hauk, O., Grave de Peralta Menendez, R. and Lütkenhöner, B. (1996): The Backus and Gilbert method and the minimum norm method applied to a simple model of a cortex fold. Proceedings of the Third International Hans Berger Congress, Jena, Germany (to appear).

    Google Scholar 

  • Hämäläinen, M.S. and Ilmoniemi, R.J. (1984): Interpreting measured magnetic fields of the brain: Estimates of current distributions. Technical Report TKK- F-A559, Helsinski University of Technology.

    Google Scholar 

  • Hämäläinen, M.S., Hari, R., Ilmoniemi, R.J., Knuutila, J. and Lounasma, O.V. (1993): Magnetoencephalography — theory, instrumentation, and applications to non invasive studies of the working human brain. Rev. Mod. Phys. 65:413–497. 1

    Google Scholar 

  • Ioannides, A.A., Bolton, J.P.R., Hasson, R. and Clarke C.J.S. (1989): Localised and distributed source solutions for the biomagnetic inverse problem II. In: Williamson S J, Hoke M, Stroink G and Kotani M (eds): Advances in Biomagnetism. New York: Plenum Press, pp. 587 – 590.

    Google Scholar 

  • Knösche, T. (1997): Three-dimensional reconstruction from EEG, the performance of linear algoritmhs. Biomedizinische Technik, 42: 205 – 208.

    Article  Google Scholar 

  • Lanz, G., Michel, C.M., Pascual Marqui, R.D., Spinelli, L., Seeck, M., Seri, S., Landis, T. and Rosem, I. (1997) Extracranial localization of intracranial interictal epileptiform activity using LORETA (low resolution electromagnetic tomography). Electroenc. Clin. Neurophysiol., 102: 414 – 422.

    Google Scholar 

  • Lawson C.L. and Hanson, R.J. (1974): Solving least squares problems. (Prentice Hall, Inc., Englewood Cliffs, New Jersey).

    MATH  Google Scholar 

  • Leahy, R., Mosher, J.C., and Phillips, J.W. (1996): A comparative study of minimum norm inverse methods for MEG imaging. Proceedings of the tenth inter¬national conference on Biomagnetism, Biomag ’96. Santa Fe, New Mexico

    Google Scholar 

  • Linz, P. (1979): Theoretical numerical analysis. An introduction to advanced techniques. (John Wiley & Sons, Inc., New York).

    MATH  Google Scholar 

  • Matsuura, K. and Okabe, Y. (1995): Selective minimum norm solution of the biomagnetic inverse problem. IEEE Trans. Biomed. Engn., 42: 608 – 615.

    Article  Google Scholar 

  • Menke, W. (1989): Geophysical Data Analysis: Discrete inverse theory. Academic Press, San Diego, California.

    MATH  Google Scholar 

  • Mosher, J.C., Lewis, P.S. and Leahy, L. (1992): Multiple dipole modeling and localization from spatio temporal MEG data. IEEE Trans. Biomed. Eng., 39: 541 – 557.

    Article  Google Scholar 

  • Okada, Y., Huang, J., and Xu, C. (1992): A hierarchical minimum norm estimation method for reconstructing current densities in the brain from remotely measured magnetic fields, in: Biomagnetism: Clinical Aspects, edited by Hoke, M., Erne, S., Okada, Y. and Romani, G.L. Elsevier. Amsterdam, pp. 729 – 734.

    Google Scholar 

  • Pascual Marqui, R.D. and Michel, C.M. (1994): Rejoinder. In: ISBET Newsletter No. 5, November 1994. Ed: W. Skrandies. 21 – 25.

    Google Scholar 

  • Pascual Marqui, R.D. (1995): Reply to comments by Hämäläinen, Ilmoniemi and Nunez. In: ISBET Newsletter No. 6, December 1995. Ed: W. Skrandies. 16 – 28.

    Google Scholar 

  • Pascual Marqui, R.D., Michel, C.M. and Lehmann D. (1995): Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain. Int. J. Psychophysiol. 18: 49 – 65.

    Article  Google Scholar 

  • Penrose R. (1955): A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51: 406 – 413.

    MathSciNet  MATH  Google Scholar 

  • Phillips, J.W., Leahy, R., and Mosher, J.C. (1997): MEG-based imaging of focal neuronal sources, IEEE Trans, on Medical Imaging 16: 338 – 348

    Article  Google Scholar 

  • Plonsey, R. (1982). The nature of the sources of bioelectric and biomagnetic fields. Biophys. J. 39. pp. 309 – 312.

    Article  Google Scholar 

  • Rao, C.R. and Mitra, S.K. (1971): Generalized inverse of matrices and its applica¬tions. (John Wiley & Sons, Inc., New York).

    Google Scholar 

  • Riera, J.J., Valdes, P.A., Fuentes, M.A. and Oharriz, Y. (1997a): Explicit Backus and Gilbert EEG Inverse Solutions for a Spherical Symmetry. Biomedizinische Technik, 42: 216 – 219

    Article  Google Scholar 

  • Riera, J.J., Fuentes, M.A., Valdes, P.A. and Oharriz, Y. (1997b): Theoretical basis of the EEG Spline Inverse Solutions for a spherical Head Model. Biomedizinische Technik, 42: 219 – 223

    Article  Google Scholar 

  • Roach, G.F. (1970): Green functions. Introductory theory with applications. Van Nostrand Reinhold Co.

    Google Scholar 

  • Sarvas, J. (1987): Basic mathematical and electromagnetic concepts of the bioelectromagnetic inverse problem. Phys. Med. Biol. 32: 11 – 22.

    Google Scholar 

  • Scherg, M., von Cramon, D. (1990): Dipole source potentials of the auditory cortex in normal subjects and patients with temporal lobe lesions. In: Grandori, F., Hoke, M., Romani, G.L., eds. Auditory evoked magnetic fields and electric potentials. ( Karger, Basel)., 165 – 193.

    Google Scholar 

  • Sekihara, K. and Scholz, B. (1995): Average-intensity reconstruction and Wiener re-construction of bioelectric current distribution based on its estimated covariance matrix. IEEE Trans. Biomed. Eng. 42: 149 – 157.

    Article  Google Scholar 

  • Sekihara, K. and Scholz, B. (1996): Generalized wiener estimation of three dimensional current distribution from biomagnetic measurements. IEEE Trans. Biomed. Eng. 43: 281 – 291.

    Article  Google Scholar 

  • Sekihara, K., Poeppel, D., Marantz, A., Koizumi, H., Miyashita, Y. (1997a): Noise covariance incorporated MEG-Music algorithms: A method for multiple dipole estimation tolerant of the influence of Background Brain activity. IEEE Trans. Biomed. Eng. Accepted for publication.

    Google Scholar 

  • Sekihara, K., Poeppel, D., Marantz, A., Philips, C., Koizumi, H., Miyashita, Y. (1997b): MEG Covariance difference analysis: A method to extract target source activities by using task and control measurements. Preprint.

    Google Scholar 

  • Sezan, M.I., Stark, H. (1982): Image restoration by the method of convex projections: Part II-Application and Numerical results. IEEE Trans. Med. Imaging, vol Ml-1: 95 – 101.

    Google Scholar 

  • Shimogawara, M.K. and Higuchi, H.M. (1992): Magnetic source imaging by current element distribution, in Biomagnetism: Clinical Aspects, edited by Hoke, M., Erne, S., Okada, Y. and Romani, G.L. Elsevier. Amsterdam, pp. 757 – 760.

    Google Scholar 

  • Simard, P.Y. and Mailloux, G.E. (1990): Vector field restoration by the method of convex projection. Computer Vision., Graphics and Image Processing, 52: 360 – 385.

    Article  Google Scholar 

  • Spinelli, L., Pascual Marqui, R., Grave de Peralta Menendez, R., and Michel, C.M. (1997): Effect of the number and the configuration of electrodes on distributed source models. VIII World Congress of the ISBET. Zurich, Switzerland.

    Google Scholar 

  • Srebro, R. (1996): An iterative approach to the solution of the inverse problem. Electroenceph. clin. Neurophysiol., 98: 349 – 362.

    Google Scholar 

  • Tarantola, A. (1987): Inverse Problem Theory. Elsevier.

    Google Scholar 

  • Valdes, P.A., Grave de Peralta Menendez, R. and Gonzalez Andino, S.L., (1994): Comments on LORETA. In: ISBET Newsletter No. 5, November 1994. Ed: W. Skrandies. 18 – 21.

    Google Scholar 

  • Wahba, G. (1990): Spline models for observational data. Society for Industrial and applied mathematics. Philadelphia. Pennsylvania.

    Google Scholar 

  • Wagner, M. Fuchs, H.-A. Wischmann, R. Drenckhahn, Th. Köhler (1996): Current Density Reconstructions Using the LÍ Norm. Proc. of the 10th Int. Conf. of Biomagnetism; BIOMAG 96, Santa Fe, 16. – 21.2.1996, to be published (Abstracts, pp. 168 ).

    Google Scholar 

  • Wikswo J.P., Jr., Malmivuo, J.A.V., Barry, W.H., Leifer, M.C. and Fairbank W.M. (1979): The theory and application of magnetocardiography. Adv. Cardiovasc. Phys, 2: 1 – 67.

    Google Scholar 

  • Youla, D.C. and Webb, H. (1982): Image restoration by the method of convex projections: Part I-Theory. IEEE Trans. Med. Imaging, vol Ml-1: 81 – 94.

    Google Scholar 

  • Ivert B. (1996): Modélisation réaliste de l’activité électrique cérébrale. Ph.D. thesis.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Functional Brain Mapping Lab., Dept. of Neurology, Geneva University Hospital, 1211, Geneva 14, Switzerland

    Rolando Grave de Peralta Menendez & Sara Gonzalez Andino

Authors
  1. Rolando Grave de Peralta Menendez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sara Gonzalez Andino
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Max-Planck-Institute of Cognitive Neuroscience, D-04103, Stephanstrasse 1a, Leipzig, Germany

    Christian Uhl

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Grave de Peralta Menendez, R., Andino, S.G. (1999). Distributed Source Models: Standard Solutions and New Developments. In: Uhl, C. (eds) Analysis of Neurophysiological Brain Functioning. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60007-4_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-60007-4_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64219-7

  • Online ISBN: 978-3-642-60007-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation